外文翻译

Strengths优势All these private sector banks hold strong position on CRM part, they have professional, dedicated and well-trained employees.所以这些私人银行在客户管理部分都持支持态度，他们拥有专业的、细致的、训练有素的员工。

Private sector banks offer a wide range of banking and financial products and financial services to corporate and retail customers through a variety of delivery channels such as ATMs, Internet-banking, mobile-banking, etc. 私有银行通过许多传递通道（如自动取款机、网上银行、手机银行等）提供大范围的银行和金融产品、金融服务进行合作并向客户零售。

The area could be Investment management banking, life and non-life insurance, venture capital and asset management, retail loans such as home loans, personal loans, educational loans, car loans, consumer durable loans, credit cards, etc. 涉及的领域包括投资管理银行、生命和非生命保险、风险投资与资产管理、零售贷款（如家庭贷款、个人贷款、教育贷款、汽车贷款、耐用消费品贷款、信用卡等）。

Private sector banks focus on customization of products that are designed to meet the specific needs of customers. 私人银行主要致力于为一些特殊需求的客户进行设计和产品定制。

文献综述外文翻译写作规范及要求
文献综述是对已经发表的学术文献进行系统的综合分析和评价的一种
学术写作形式。

在撰写文献综述的过程中，外文翻译是不可或缺的一部分。

下面是一些关于外文翻译的写作规范和要求。

1.准确：外文翻译要准确无误地表达原文的意思。

翻译过程中应注意
遵守语法规则、掌握专业术语以及正确理解上下文。

2.逻辑清晰：翻译后的中文句子要符合中文语法和表达习惯，并保持
逻辑上的连贯。

避免使用过于生硬或拗口的句子结构。

3.简洁明了：文献综述注重对已有文献的概括和总结，因此翻译过程
中应力求简洁明了，避免翻译过多的细节和废话。

4.专业术语准确翻译：外文翻译中的专业术语在翻译过程中要保持准
确性。

可以参考已有的专业词典、论文翻译表格等工具来确保专业术语的
正确翻译。

5.文体和语气恰当：根据不同的文献类型和句子语境，选择合适的文
体和语气进行翻译。

可以参考论文综述的写作规范和范例，避免过于口语
化或过于正式的翻译。

在撰写文献综述的过程中，准确和恰当的外文翻译是非常重要的。

只
有通过准确和规范的翻译，才能保证文献综述的质量和可信度。

因此，应
该注重提升外文翻译的能力并积极学习相关的写作规范和要求。

因为学校对毕业论文中的外文翻译并无规定，为统一起见,特做以下要求：1、每篇字数为1500字左右，共两篇；2、每篇由两部分组成:译文+原文.3 附件中是一篇范本，具体字号、字体已标注。

外文翻译（包含原文)(宋体四号加粗）外文翻译一（宋体四号加粗）作者：(宋体小四号加粗）Kim Mee Hyun Director, Policy Research & Development Team，Korean Film Council（小四号）出处：(宋体小四号加粗)Korean Cinema from Origins to Renaissance(P358～P340) 韩国电影的发展及前景（标题：宋体四号加粗）1996～现在数量上的增长(正文:宋体小四）在过去的十年间，韩国电影经历了难以置信的增长。

上个世纪60年代，韩国电影迅速崛起，然而很快便陷入停滞状态,直到90年代以后，韩国电影又重新进入繁盛时期。

在这个时期，韩国电影在数量上并没有大幅的增长，但多部电影的观影人数达到了上千万人次。

1996年，韩国本土电影的市场占有量只有23.1%。

但是到了1998年，市场占有量增长到35。

8%，到2001年更是达到了50%。

虽然从1996年开始，韩国电影一直处在不断上升的过程中，但是直到1999年姜帝圭导演的《生死谍变》的成功才诞生了韩国电影的又一个高峰。

虽然《生死谍变》创造了韩国电影史上的最高电影票房纪录，但是1999年以后最高票房纪录几乎每年都会被刷新。

当人们都在津津乐道所谓的“韩国大片”时，2000年朴赞郁导演的《共同警备区JSA》和2001年郭暻泽导演的《朋友》均成功刷新了韩国电影最高票房纪录.2003年康佑硕导演的《实尾岛》和2004年姜帝圭导演的又一部力作《太极旗飘扬》开创了观影人数上千万人次的时代。

姜帝圭和康佑硕导演在韩国电影票房史上扮演了十分重要的角色。

从1993年的《特警冤家》到2003年的《实尾岛》，康佑硕导演了多部成功的电影。

毕业论文外文翻译格式毕业论文外文翻译格式在撰写毕业论文时，外文翻译是一个重要的环节。

无论是引用外文文献还是翻译相关内容，都需要遵循一定的格式和规范。

本文将介绍一些常见的外文翻译格式，并探讨其重要性和应用。

首先，对于引用外文文献的格式，最常见的是使用APA（American Psychological Association）格式。

这种格式要求在引用外文文献时，先列出作者的姓氏和名字的首字母，然后是出版年份、文章标题、期刊名称、卷号和页码。

例如：Smith, J. D. (2010). The impact of climate change on biodiversity. Environmental Science, 15(2), 145-156.在翻译外文文献时，需要注意保持原文的准确性和完整性。

尽量避免意译或添加自己的解释，以免歪曲原文的意思。

同时，还需要在翻译后的文献后面加上“译者”和“翻译日期”的信息，以便读者可以追溯翻译的来源和时间。

其次，对于翻译相关内容的格式，可以参考国际标准组织ISO（International Organization for Standardization）的格式。

这种格式要求在翻译相关内容时，先列出原文，然后是翻译后的文本。

例如：原文：The importance of effective communication in the workplace cannot be overstated.翻译：工作场所有效沟通的重要性不容忽视。

在翻译相关内容时，需要注意保持原文的意思和语气。

尽量使用准确的词汇和语法结构，以便读者能够理解和接受翻译后的内容。

同时，还需要在翻译后的文本后面加上“翻译者”和“翻译日期”的信息，以便读者可以追溯翻译的来源和时间。

此外，对于长篇外文文献的翻译，可以考虑将其分成若干章节，并在每个章节前面加上章节标题。

这样可以使读者更容易理解和阅读翻译后的内容。

外文翻译器外文翻译器外文翻译器（Machine Translation）是指使用计算机等技术对外文进行自动翻译的工具。

它利用计算机语言处理、人工智能和语言学等多个领域的知识和技术，将源语言（外文）自动转化为目标语言（母语）的过程。

外文翻译器可以帮助人们快速准确地将外文内容转化为自己熟悉的语言，提高工作效率和信息获取能力。

外文翻译器的研究和发展始于上世纪40年代，最早采用的是基于规则的翻译方法，即根据语法规则和词汇库对源语言进行分析和转换。

然而，这种方法存在很多限制，因为语法和词汇库可能无法覆盖所有的语言特点和用法，导致翻译结果不准确和不流畅。

随着计算机技术和人工智能的发展，神经网络机器翻译（Neural Network Translation）成为外文翻译器的主流方法。

这种方法利用大规模平行语料库训练神经网络模型，通过模仿人类学习语言的方式自动学习源语言和目标语言之间的映射关系。

神经网络机器翻译能够更好地处理语法结构和上下文信息，翻译结果更加准确和自然。

除了神经网络机器翻译，外文翻译器还可以采用统计机器翻译（Statistical Machine Translation）等其他方法。

统计机器翻译利用大量的双语语料进行统计分析，找到最佳的翻译候选，然后根据概率模型对其进行排序和选择。

虽然统计机器翻译在一定程度上改善了翻译质量，但由于依赖于大量的语料库，对于某些语言和领域的翻译效果仍然不理想。

当前外文翻译器的发展已经进入了深度学习时代，融合了自然语言处理、深度学习和人工智能的多种技术手段。

深度学习通过建立多层神经网络模型，能够从大规模语料中自动学习和提取特征，进一步提升了翻译质量和效率。

此外，人工智能的发展还带来了一系列辅助工具，如术语提取、句子结构分析和语音识别等，能够进一步提高翻译的准确性和流畅度。

虽然外文翻译器在很大程度上改善了翻译效率和准确性，但由于语言本身的复杂性和多义性，完全依靠机器翻译仍然存在一些局限性。

毕业外文翻译Graduation is an important milestone in one's life. It signifies the completion of a period of education and the beginning of a new chapter. It is a time to reflect on the memories and experiences gained during our time in school.During the graduation ceremony, students are filled with a mix of emotions. There is a sense of accomplishment and pride for having successfully completed the course of study. There may also be a feeling of nostalgia as we bid farewell to the friends and teachers who have been a significant part of our lives. Graduation is a time to celebrate not only the academic achievements, but also the personal growth and development that has taken place over the years.For many students, graduation is both a time of excitement and uncertainty. It is a time to look forward to the future, but also a time to face the challenges and uncertainties that come with it. It is a time to reflect on our goals and dreams, and to make plans for the years to come. Graduation marks the transition from the structured environment of school to the more independent and unpredictable world beyond. It is a time to step out of our comfort zones and embrace new opportunities and experiences.Graduation is not only a significant event for students, but also for their families and loved ones. They have supported us throughout our educational journey, providing encouragement, guidance, and love. Graduation is a time for them to share in our achievements and to feel proud of our accomplishments. It is a time to express our gratitude to those who have been with us every step of the way.Beyond the celebrations and pomp and circumstance, graduation is also a time for reflection and self-evaluation. It is a time to look back on the challenges we have overcome, the lessons we have learned, and the growth we have experienced. It is a time to set new goals and aspirations, and to make a commitment to continue learning and growing.In conclusion, graduation is a momentous occasion that marks the culmination of years of hard work and dedication. It is a time to celebrate our accomplishments, reflect on our experiences, and look forward to the future. Graduation is not only an individual achievement, but also a celebration of the support and love of our families and friends. It is a time to embrace the unknown and embark on new adventures. Graduation is not the end, but rather the beginning of a new journey.。

外文翻译规范要求及模版格式
外文中文翻译规范要求及模板格式可以根据具体需求和要求有所不同，以下是一般常见的外文中文翻译规范要求及模板格式：
1.规范要求：
-符合语法、语言规范和语义准确性；
-译文流畅自然，符合中文表达习惯；
-忠实准确地传达原文信息；
-注意统一使用特定的术语翻译；
-文章结构、段落、标题等要与原文一致；
-保持适当的篇幅，不过度增加或删减内容；
-遵守保密原则。

2.模板格式：
-文章标题（与原文保持一致，可放在正文上方）；
-标题（与原文保持一致）；
-段落（与原文保持一致，首行缩进）；
-字体（常用宋体或黑体，一般字号12或14）；
-行间距（一般1.5倍，可根据需要调整）；
-页边距（上下左右均为2.5厘米）；
-段落间距（一般1.5倍，可根据需要调整）；
以上是一般常见的外文中文翻译规范要求及模板格式，具体要求和格式可以根据具体的翻译项目和要求进行调整。

在翻译过程中，保持准确、流畅、专业是非常重要的。

论文外文翻译指导日志
翻译要求：
1、选定外文文献后先给指导老师看，得到老师的确认通过后方可翻译。

2、选择外文翻译时一定选择外国作者写的文章，可从学校中知网或者外文数据库下载。

3、外文翻译字数要求3000字以上，从外文文章起始处开始翻译，不允许从文章中间部分开始翻译，翻译必须结束于文章的一个大段落。

参考文献是在学术研究过程中，对某一著作或论文的整体的参考或借鉴。

征引过的文献在注释中已注明,不再出现于文后参考文献中。

外文参考文献就是指论文是引用的文献原文是国外的，并非中国的。

原文就是指原作品,原件，即作者所写作品所用的语言。

如莎士比亚的《罗密欧与朱丽叶》原文是英语。

译文就是翻译过来的文字，如在中国也可以找到莎士比亚《罗密欧与朱丽叶》的中文版本，这个中文版本就称为译文。

主要标准
翻译是语际交流过程中沟通不同语言的桥梁。

一般来说，翻译的标准主要有两条：忠实和通顺。

忠实
是指忠实于原文所要传递的信息，也就是说，把原文的信息完整并且准确地表达出来，使译文读者得到的信息与原文读者得到的信息大致相同。

通顺
是指译文规范、明白易懂，没有文理不通、结构混乱、逻辑不清的现象。

外文翻译范例在全球化日益加深的今天，外文翻译的重要性愈发凸显。

无论是学术研究、商务交流，还是文化传播，准确而流畅的外文翻译都起着至关重要的桥梁作用。

下面为大家呈现几个不同领域的外文翻译范例，以帮助大家更好地理解和掌握外文翻译的技巧与要点。

一、科技文献翻译原文：The development of artificial intelligence has brought about revolutionary changes in various fields, such as healthcare, finance, and transportation译文：人工智能的发展给医疗保健、金融和交通运输等各个领域带来了革命性的变化。

在这个范例中，翻译准确地传达了原文的意思。

“artificial intelligence”被准确地翻译为“人工智能”，“revolutionary changes”翻译为“革命性的变化”，“various fields”翻译为“各个领域”，用词准确、贴切，符合科技文献严谨、客观的语言风格。

二、商务合同翻译原文：This Agreement shall commence on the effective date and shall continue in force for a period of five years, unless earlier terminated in accordance with the provisions herein译文：本协议自生效日起生效，并将持续有效五年，除非根据本协议的规定提前终止。

商务合同的翻译需要格外注重准确性和专业性。

上述译文中，“commence”翻译为“生效”，“in force”翻译为“有效”，“terminated”翻译为“终止”，清晰准确地表达了合同条款的含义，避免了可能的歧义。

三、文学作品翻译原文：The sun was setting, painting the sky with hues of orange and pink, as if nature were a master artist at work译文：太阳正在西沉，把天空涂成了橙色和粉色，仿佛大自然是一位正在创作的艺术大师。

DOI10.1007/s10711-012-9699-zORIGINAL PAPERParking garages with optimal dynamicsMeital Cohen·Barak WeissReceived:19January2011/Accepted:22January2012©Springer Science+Business Media B.V.2012Abstract We construct generalized polygons(‘parking garages’)in which the billiard flow satisfies the Veech dichotomy,although the associated translation surface obtained from the Zemlyakov–Katok unfolding is not a lattice surface.We also explain the difficulties in constructing a genuine polygon with these properties.Keywords Active vitamin D·Parathyroid hormone-related peptide·Translation surfaces·Parking garages·Veech dichotomy·BilliardsMathematics Subject Classification(2000)37E351Introduction and statement of resultsA parking garage is an immersion h:N→R2,where N is a two dimensional compact connected manifold with boundary,and h(∂N)is afinite union of linear segments.A parking garage is called rational if the group generated by the linear parts of the reflections in the boundary segments isfinite.If h is actually an embedding,the parking garage is a polygon; thus polygons form a subset of parking garages,and rationals polygons(i.e.polygons all of whose angles are rational multiples ofπ)form a subset of rational parking garages.The dynamics of the billiardflow in a rational polygon has been intensively studied for over a century;see[7]for an early example,and[5,10,13,16]for recent surveys.The defi-nition of the billiardflow on a polygon readily extends to a parking garage:on the interior of N the billiardflow is the geodesicflow on the unit tangent bundle of N(with respect to the pullback of the Euclidean metric)and at the boundary,theflow is defined by elastic reflection (angle of incidence equals the angle of return).Theflow is undefined at thefinitely many M.Cohen·B.Weiss(B)Ben Gurion University,84105Be’er Sheva,Israele-mail:barakw@math.bgu.ac.ilM.Cohene-mail:comei@bgu.ac.ilpoints of N which map to‘corners’,i.e.endpoints of boundary segments,and hence at thecountable union of codimension1submanifolds corresponding to points in the unit tangentbundle for which the corresponding geodesics eventually arrive at corners in positive or neg-ative time.Since the direction of motion of a trajectory changes at a boundary segment viaa reflection in its side,for rational parking garages,onlyfinitely many directions of motionare assumed.In other words,the phase space of the billiardflow decomposes into invarianttwo-dimensional subsets corresponding tofixing the directions of motion.Veech[12]discovered that the billiardflow in some special polygons exhibits a strikingly he found polygons for which,in any initial direction,theflow is eithercompletely periodic(all orbits are periodic),or uniquely ergodic(all orbits are equidistrib-uted).Following McMullen we will say that a polygon with these properties has optimaldynamics.We briefly summarize Veech’s strategy of proof.A standard unfolding construc-tion usually attributed to Zemlyakov and Katok[15]1,associates to any rational polygon Pa translation surface M P,such that the billiardflow on P is essentially equivalent to thestraightlineflow on M P.Associated with any translation surface M is a Fuchsian group M,now known as the Veech group of M,which is typically trivial.Veech found M and P forwhich this group is a non-arithmetic lattice in SL2(R).We will call these lattice surfaces and lattice polygons respectively.Veech investigated the SL2(R)-action on the moduli space of translation surfaces,and building on earlier work of Masur,showed that lattice surfaces haveoptimal dynamics.From this it follows that lattice polygons have optimal dynamics.This chain of reasoning remains valid if one starts with a parking garage instead of apolygon;namely,the unfolding construction associates a translation surface to a parkinggarage,and one may define a lattice parking garage in an analogous way.The arguments ofVeech then show that the billiardflow in a lattice parking garage has optimal dynamics.Thisgeneralization is not vacuous:lattice parking garages,which are not polygons,were recentlydiscovered by Bouw and Möller[2].The term‘parking garage’was coined by Möller.A natural question is whether Veech’s result admits a converse,i.e.whether non-latticepolygons or parking garages may also have optimal dynamics.In[11],Smillie and the sec-ond-named author showed that there are non-lattice translation surfaces which have optimaldynamics.However translation surfaces arising from billiards form a set of measure zero inthe moduli space of translation surfaces,and it was not clear whether the examples of[11]arise from polygons or parking garages.In this paper we show:Theorem1.1There are non-lattice parking garages with optimal dynamics.An example of such a parking garage is shown in Fig.1.Veech’s work shows that for lattice polygons,the directions in which all orbits are periodicare precisely those containing a saddle connection,i.e.a billiard path connecting corners ofthe polygon which unfold to singularities of the corresponding surface.Following Cheunget al.[3],if a polygon P has optimal dynamics,and the periodic directions coincide with thedirections of saddle connections,we will say that P satisfies strict ergodicity and topologicaldichotomy.It is not clear to us whether our example satisfies this stronger property.As weexplain in Remark3.2below,this would follow if it were known that the center of the regularn-gon is a‘connection point’in the sense of Gutkin,Hubert and Schmidt[8]for some nwhich is an odd multiple of3.Veech also showed that for a lattice polygon P,the number N P(T)of periodic strips on P of length at most T satisfies a quadratic growth estimate of the form N P(T)∼cT2for a positive constant c.As we explain in Remark3.3,our examples also satisfy such a quadratic growth estimate.1But dating back at least to Fox and Kershner[7].Fig.1A non-lattice parkinggarage with optimal dynamics.(Here 2/n represents angle 2π/n )It remains an open question whether there is a genuine polygon which has optimal dynam-ics and is not a lattice polygon.Although our results make it seem likely that such a polygon exists,in her M.Sc.thesis [4],the first-named author obtained severe restrictions on such a polygon.In particular she showed that there are no such polygons which may be constructed from any of the currently known lattice examples via the covering construction as in [11,13].We explain these results and prove a representative special case in §4.2PreliminariesIn this section we cite some results which we will need,and deduce simple consequences.For the sake of brevity we will refer the reader to [10,11,16]for definitions of translation surfaces.Suppose S 1,S 2are compact orientable surfaces and π:S 2→S 1is a branched cover.That is,πis continuous and surjective,and there is a finite 1⊂S 1,called the set of branch points ,such that for 2=π−1( 1),the restriction of πto S 2 2is a covering map of finite degree d ,and for any p ∈ 1,#π−1(p )<d .A ramification point is a point q ∈ 2for which there is a neighborhood U such that {q }=U ∩π−1(π(q ))and for all u ∈U {q },# U ∩π−1(π(u )) ≥2.If M 1,M 2are translation surfaces,a translation map is a surjective map M 2→M 1which is a translation in charts.It is a branched cover.In contrast to other authors (cf.[8,13]),we do not require that the set of branch points be distinct from the singularities of M 1,or that they be marked.It is clear that the ramification points of the cover are singularities on M 2.If M is a lattice surface,a point p ∈M is called periodic if its orbit under the group of affine automorphisms of M is finite.A point p ∈M is called a connection point if any seg-ment joining a singularity with p is contained in a saddle connection (i.e.a segment joining singularities)on M .The following proposition summarizes results discussed in [7,9–11]:Proposition 2.1(a)A non-minimal direction on a translation surface contains a saddle connection.(b)If M 1is a lattice surface,M 2→M 1is translation map with a unique branch point,then any minimal direction on M 2is uniquely ergodic.(c)If M2→M1is a translation map such that M1is a lattice surface,then all branchpoints are periodic if and only if M2is a lattice surface.(d)If M2→M1is a translation map with a unique branch point,such that M1is a latticesurface and the branch point is a connection point,then any saddle connection direction on M2is periodic.Corollary2.2Let M2→M1be a translation map such that M1is a lattice surface with a unique branch point p.Then:(1)M2has optimal dynamics.(2)If p is a connection point then M2satisfies topological dichotomy and strict ergodicity.(3)If p is not a periodic point then M2is not a lattice surface.Proof To prove(1),by(b),the minimal directions are uniquely ergodic,and we need to prove that the remaining directions are either completely periodic or uniquely ergodic. By(a),in any non-minimal direction on M2there is a saddle connectionδ,and there are three possibilities:(i)δprojects to a saddle connection on M1.(ii)δprojects to a geodesic segment connecting the branch point p to itself.(iii)δprojects to a geodesic segment connecting p to a singularity.In case(i)and(ii)since M1is a lattice surface,the direction is periodic on M1,hence on M2as well.In case(iii),there are two subcases:ifδprojects to a part of a saddle connec-tion on M1,then it is also a periodic direction.Otherwise,in light of Proposition2.1(a),the direction must be minimal in M1,and hence,by Proposition2.1(b),uniquely ergodic in M2. This proves(1).Note also that if p is a connection point then the last subcase does not arise, so all directions which are non-minimal on M2are periodic.This proves(2).Statement(3) follows from(c).We now describe the unfolding construction[7,15],extended to parking garages.Let P=(h:N→R2).An edge of P is a connected subset L of∂N such that h(L)is a straight segment and L is maximal with these properties(with respect to inclusion).A vertex of P is any point which is an endpoint of an edge.The angle at a vertex is the total interior angle, measured via the pullback of the Euclidean metric,at the vertex.By convention we always choose the positive angles.Note that for polygons,angles are less than2π,but for parking garages there is no apriori upper bound on the angle at a vertex.Since our parking garages are rational,all angles are rational multiples ofπ,and we always write them as p/q,omitting πfrom the notation.Let G P be the dihedral group generated by the linear parts of reflections in h(L),for all edges L.For the sake of brevity,if there is a reflection with linear part gfixing a line parallel to L,we will say that gfixes L.Let S be the topological space obtained from N×G P by identifying(x,g1)with(x,g2)whenever g−11g2fixes an edge containing h(x).Topologically S is a compact orientable surface,and the immersions g◦h on each N×{g}induce an atlas of charts to R2which endows S with a translation surface structure.We denote this translation surface by M P,and writeπP for the map N×G P→M P.We will be interested in a‘partial unfolding’which is a variant of this construction,in which we reflect a parking garage repeatedly around several of its edges to form a larger parking garage.Formally,suppose P=(h:N→R2)and Q=(h :N →R2)are parking garages.For ≥1,we say that P tiles Q by reflections,and that is the number of tiles,if the following holds.There are maps h 1,...h :N→N and g1,...,g ∈G P(not necessarily distinct)satisfying:(A)The h i are homeomorphisms onto their images,and N = h i (N ).(B)For each i ,the linear part of h ◦h i ◦h −1is everywhere equal to g i .(C)For each 1≤i <j ≤ ,let L i j =h i (N )∩h j (N )and L =(h i )−1(L i j ).Then (h j )−1◦h i is the identity on L ,and L is either empty,or a vertex,or an edge of P .If L is an edge then h i (N )∪h j (N )is a neighborhood of L i j.If L i j is a vertex then there is a finite set of i =i 1,i 2,...,i k =j such that h i s (N )contains a neighborhood of L i j ,and each consecutive pair h i t (N ),h i t +1(N )intersect along an edge containing L i j .V orobets [13]realized that a tiling of parking garages gives rise to a branched cover.More precisely:Proposition 2.3Suppose P tiles Q by reflections with tiles,M P ,M Q are the correspond-ing translation surfaces obtained via the unfolding construction,and G P ,G Q are the cor-responding reflection groups.Then there is a translation map M Q →M P ,such that the following hold:(1)G Q ⊂G P .(2)The branch points are contained in the G P -orbit of the vertices of P .(3)The degree of the cover is [G P :G Q ].(4)Let z ∈M P be a point which is represented (as an element of N ×{1,...,r })by(x ,k )with x a vertex in P with angle m n (where gcd (m ,n )=1).Let (y i )⊂M Q be the pre-images of z,with angles k i m n in Q .Then z is a branch point of the cover if and only if k i n for some i.Proof Assertion (1)follows from the fact that Q is tiled by P .Since this will be impor-tant in the sequel,we will describe the covering map M Q →M P in detail.We will map (x ,g )∈N ×G Q to πP (x ,gg i )∈M P ,where x =h i (x ).We now check that this map is independent of the choice of x ,i ,and descends to a well-defined map M Q →M P ,which is a translation in charts.If x =h i (x 1)=h j (x 2)then x 1=x 2since (h i )−1◦h j is the identity.If x is in the relative interior of an edge L i j thenπP (x ,gg i )=πP (x ,gg j )(1)since (gg i )−1gg j =g −1i g j fixes an edge containing h (x 1).If x 1is a vertex of P then one proves (1)by an induction on k ,where k is as in (C).This shows that the map is well-defined.We now show that it descends to a map M Q →M P .Suppose (x ,g ),(x ,g )are two points in N ×G Q which are identified in M Q ,i.e.x ∈∂N is in the relative interior of an edge fixed by g −1g .By (C)there is a unique i such that x is in the image of h i .Thus (x ,g )maps to (x ,gg i )and (x ,g )maps to (x ,g g i ),and g −1i g −1g g i fixes the edge through x =g −1i (x ).It remains to show that the map we have defined is a translation in charts.This follows immediately from the chain rule and (B).Assertion (2)is simple and left to the reader.For assertion (3)we note that M P (resp.M Q )is made of |G P |(resp. |G Q |)copies of P .The point z will be a branch point if and only if the total angle around z ∈M P differs from the total angle around one of the pre-images y i ∈M Q .The total angle at a singularity corresponding to a vertex with angle r /s (where gcd (r ,s )=1)is 2r π,thus the total angle at z is 2m πand the total angle at y i is 2k i m πgcd (k i ,n ).Assertion (4)follows.3Non-lattice dynamically optimal parking garagesIn this section we prove the following result,which immediately implies Theorem1.1: Theorem3.1Let n≥9be an odd number divisible by3,and let P be an isosceles triangle with equal angles1/n.Let Q be the parking garage made of four copies of P glued as in Fig.1, so that Q has vertices(in cyclic order)with angles1/n,2/n,3/n,(n−2)/n,2/n,3(n−2)/n. Then M P is a lattice surface and M Q→M P is a translation map with one aperiodic branchpoint.In particular Q is a non-lattice parking garage with optimal dynamics.Proof The translation surface M P is the double n-gon,one of Veech’s original examples of lattice surfaces[12].The groups G P and G Q are both equal to the dihedral group D n.Thus by Proposition2.3,the degree of the cover M Q→M P is four.Again by Proposition2.3, since n is odd and divisible by3,the only vertices which correspond to branch points are the two vertices z1,z2with angle2/n(they correspond to the case k i=2while the other vertices correspond to1or3).In the surface M P there are two points which correspond to vertices of equal angle in P(the centers of the two n-gons),and these points are known to be aperiodic [9].We need to check that z1and z2both map to the same point in M P.This follows from the fact that both are opposite the vertex z3with angle3/n,which also corresponds to the center of an n-gon,so in M P project to a point which is distinct from z3. Remark3.2As of this writing,it is not known whether the center of the regular n-gon is a connection point on the double n-gon surface.If this turns out to be the case for some n which is an odd multiple of3,then by Corollary2.2(2),our construction satisfies strict ergodicity and topological dichotomy.See[1]for some recent related results.Remark3.3Since our examples are obtained by taking branched covers over lattice surfaces, a theorem of Eskin et al.[6,Thm.8.12]shows that our examples also satisfy a quadratic growth estimate of the form N P(T)∼cT2;moreover§9of[6]explains how one may explicitly compute the constant c.4Non-lattice optimal polygons are hard tofindIn this section we present results indicating that the above considerations will not easily yield a non-lattice polygon with optimal dynamics.Isolating the properties necessary for our proof of Theorem3.1,we say that a pair of polygons(P,Q)is suitable if the following hold:•P is a lattice polygon.•P tiles Q by reflections.•The corresponding cover M Q→M P as in Proposition2.3has a unique branch point which is aperiodic.In her M.Sc.thesis at Ben Gurion University,thefirst-named author conducted an exten-sive search for a suitable pair of polygons.By Corollary2.2,such a pair will have yielded a non-lattice polygon with optimal dynamics.The search begins with a list of candidates for P,i.e.a list of currently known lattice polygons.At present,due to work of many authors, there is a fairly large list of known lattice polygons but there is no classification of all lattice polygons.In[4],the full list of lattice polygons known as of this writing is given,and the following is proved:Theorem4.1(M.Cohen)Among the list of lattice surfaces given in[4],there is no P for which there is Q such that(P,Q)is a suitable pair.The proof of Theorem4.1contains a detailed case-by-case analysis for each of the differ-ent possible P.These cases involve some common arguments which we will illustrate in this section,by proving the special case in which P is any of the obtuse triangles investigated byWard[14]:Theorem4.2For n≥4,let P=P n be the(lattice)triangle with angles1n,12n,2n−32n.Then there is no polygon Q for which(P,Q)is a suitable pair.Our proof relies on some auxiliary statements which are of independent interest.In all of them,M Q→M P is the branched cover with unique branch point corresponding to a suitable pair(P,Q).These statements are also valid in the more general case in which P,Q are parking garages.Recall that an affine automorphism of a translation surface is a homeomorphism which is linear in charts.We denote by Aff(M)the group of affine automorphisms of M and by D:Aff(M)→GL2(R)the homomorphism mapping an affine automorphism to its linear part.Note that we allow orientation-reversing affine automorphisms,i.e.detϕmay be1 or−1.We now explain how G P acts on M P by translation equivalence.LetπP:N×G P→M P and S be as in the discussion preceding Proposition2.3,and let g∈G P.Since the left action of g on G is a permutation and preserves the gluing ruleπP,the map N×G P→N×G P sending(x,g )to(x,g−1g )induces a homeomorphismϕ:S→S and g◦h◦ϕis a translation in charts.Thus g∈G P gives a translation isomorphism of M P,and similarly g∈G P gives a translation isomorphism of M Q.Lemma4.3The branch point of the cover p:M Q→M P isfixed by G Q.Proof Since G Q⊂G P,any g∈G Q induces translation isomorphisms of both M P and M Q.We denote both by g.The definition of p given in thefirst paragraph of the proof of Proposition2.3shows that p◦g=g◦p;namely both maps are induced by sending (x ,g )∈N ×G Q toπP(x,gg g i),where x =h i(x).Since the cover p has a unique branch point,any g∈G Q mustfix it. Lemma4.4If an affine automorphismϕof a translation surface has infinitely manyfixed points then Dϕfixes a nonzero vector,in its linear action on R2.Proof Suppose by contradiction that the linear action of Dϕon the plane has zero as a uniquefixed point,and let Fϕbe the set offixed points forϕ.For any x∈Fϕwhich is not a singularity,there is a chart from a neighborhood U x of x to R2with x→0,and a smaller neighborhood V x⊂U x,such thatϕ(V x)⊂U x and when expressed in this chart,ϕ|V x is given by the linear action of Dϕon the plane.In particular x is the onlyfixed point in V x. Similarly,if x∈Fϕis a singularity,then there is a neighborhood U x of x which maps to R2 via afinite branched cover ramified at x→0,such that the action ofϕin V x⊂U x covers the linear action of Dϕ.Again we see that x is the onlyfixed point in V x.By compactness wefind that Fϕisfinite,contrary to hypothesis. Lemma4.5Suppose M is a lattice surface andϕ∈Aff(M)has Dϕ=−Id.Then afixed point forϕis periodic.Proof LetF1={σ∈Aff(M):Dσ=−Id}.Thenϕ∈F1and F1isfinite,since it is a coset for the group ker D which is known to be finite.Let A⊂M be the set of points which arefixed by someσ∈F1.By Lemma4.4this is afinite set,which contains thefixed points forϕ.Thus in order to prove the Lemma,it suffices to show that A is Aff(M)-invariant.Letψ∈Aff(M),and let x∈A,so that x=σ(x)with Dσ=−Id.Since-Id is central in GL2(R),D(σψ)=D(ψσ),so there is f∈ker D such thatψσ=fσψ.Thereforeψ(x)=ψσ(x)=fσψ(x),and fσ∈F1.This proves thatψ(x)∈A.Remark4.6This improves Theorem10of[8],where a similar conclusion is obtained under the additional assumptions that M is hyperelliptic and Aff(M)is generated by elliptic ele-ments.The following are immediate consequences:Corollary4.7Suppose(P,Q)is a suitable pair.Then•−Id/∈D(G Q).•None of the angles between two edges of Q are of the form p/q with gcd(p,q)=1and q even.Proof of Theorem4.2We will suppose that Q is such that(P,Q)are a suitable pair and reach a contradiction.If n is even,then Aff(M P)contains a rotation byπwhichfixes the points in M P coming from vertices of P.Thus by Lemma4.5all vertices of P give rise to periodic points,contradicting Proposition2.1(c).So n must be odd.Let x1,x2,x3be the vertices of P with corresponding angles1/n,1/2n,(2n−3)/2n. Then x3gives rise to a singularity,hence a periodic point.Also using Lemma4.5and the rotation byπ,one sees that x2also gives rise to a periodic point.So the unique branch point must correspond to the vertex x1.The images of the vertex x1in P give rise to two regular points in M P,marked c1,c2in Fig.2.Any element of G P acts on{c1,c2}by a permutation, so by Lemma4.3,G Q must be contained in the subgroup of index twofixing both of the c i. Let e1be the edge of P opposite x1.Since the reflection in e1,or any edge which is an image of e1under G P,swaps the c i,we have:e1is not a boundary edge of Q.(2) We now claim that in Q,any vertex which corresponds to the vertex x3from P is alwaysdoubled,i.e.consists of an angle of(2n−3)/n.Indeed,for any polygon P0,the group G P0 is the dihedral group D N where N is the least common multiple of the denominators of theangles at vertices of P0.In particular it contains-Id when N is even.Writing(2n−3)/2n in reduced form we have an even denominator,and since,by Corollary4.7,−Id/∈G Q,in Q the angle at vertex x3must be multiplied by an even integer2k.Since2k(2n−3)/2n is bigger than2if k>1,and since the total angle at a vertex of a polygon is less than2π,we must have k=1,i.e.any vertex in Q corresponding to the vertex x3is always doubled.This establishes the claim.It is here that we have used the assumption that Q is a polygon and not a parking garage.Fig.2Ward’s surface,n=5Fig.3Two options to start the construction ofQThere are two possible configurations in which a vertex x3is doubled,as shown in Fig.3. The bold lines indicate lines which are external,i.e.boundary edges of Q.By(2),the con-figuration on the right cannot occur.Let us denote the polygon on the left hand side of Fig.3by Q0.It cannot be equal to Q,since it is a lattice polygon.We now enlarge Q0by adding copies of P step by step,as described in Fig.4.Without loss of generality wefirst add triangle number1.By(2),the broken line indicates a side which must be internal in Q.Therefore,we add triangle number 2.We denote the resulting polygon by Q1.One can check by computing angles,using thefact that n is odd,and using Proposition2.3(4)that the cover M Q1→M P will branch overthe points a corresponding to vertex x2.Since the allowed branching is only over the points corresponding to x1,we must have Q1 Q,so we continue the construction.Without loss of generality we add triangle number3.Again,by(2),the broken line indicates a side which must be internal in Q.Therefore,we add triangle number4,obtaining Q2.Now,using Prop-osition2.3(4)again,in the cover M Q2→M P we have branching over two vertices u andv which are both of type x1and correspond to distinct points c1and c2in M P.This implies Q2 Q.Fig.4Steps of the construction of QSince both vertices u and v are delimited by2external sides,we cannot change the angle to prevent the branching over one of these points.This means that no matter how we continue to construct Q,the branching in the cover M Q→M P will occur over at least two points—a contradiction.Acknowledgments We are grateful to Yitwah Cheung and Patrick Hooper for helpful discussions,and to the referee for a careful reading and helpful remarks which improved the presentation.This research was supported by the Israel Science Foundation and the Binational Science Foundation.References1.Arnoux,P.,Schmidt,T.:Veech surfaces with non-periodic directions in the tracefield.J.Mod.Dyn.3(4),611–629(2009)2.Bouw,I.,Möller,M.:Teichmüller curves,triangle groups,and Lyapunov exponents.Ann.Math.172,139–185(2010)3.Cheung,Y.,Hubert,P.,Masur,H.:Topological dichotomy and strict ergodicity for translation surfaces.Ergod.Theory Dyn.Syst.28,1729–1748(2008)4.Cohen,M.:Looking for a Billiard Table which is not a Lattice Polygon but satisfies the Veech dichotomy,M.Sc.thesis,Ben-Gurion University(2010)/pdf/1011.32175.DeMarco,L.:The conformal geometry of billiards.Bull.AMS48(1),33–52(2011)6.Eskin,A.,Marklof,J.,Morris,D.:Unipotentflows on the space of branched covers of Veech surfaces.Ergod.Theorm Dyn.Syst.26(1),129–162(2006)7.Fox,R.H.,Kershner,R.B.:Concerning the transitive properties of geodesics on a rational polyhe-dron.Duke Math.J.2(1),147–150(1936)8.Gutkin,E.,Hubert,P.,Schmidt,T.:Affine diffeomorphisms of translation surfaces:Periodic points,Fuchsian groups,and arithmeticity.Ann.Sci.École Norm.Sup.(4)36,847–866(2003)9.Hubert,P.,Schmidt,T.:Infinitely generated Veech groups.Duke Math.J.123(1),49–69(2004)10.Masur,H.,Tabachnikov,S.:Rational billiards andflat structures.In:Handbook of dynamical systems,vol.1A,pp.1015–1089.North-Holland,Amsterdam(2002)11.Smillie,J.,Weiss,B.:Veech dichotomy and the lattice property.Ergod.Theorm.Dyn.Syst.28,1959–1972(2008)Geom Dedicata12.Veech,W.A.:Teichmüller curves in moduli space,Eisenstein series and an application to triangularbilliards.Invent.Math.97,553–583(1989)13.V orobets,Y.:Planar structures and billiards in rational polygons:the Veech alternative.(Russian);trans-lation in Russian Math.Surveys51(5),779–817(1996)14.Ward,C.C.:Calculation of Fuchsian groups associated to billiards in a rational triangle.Ergod.TheoryDyn.Syst.18,1019–1042(1998)15.Zemlyakov,A.,Katok,A.:Topological transitivity of billiards in polygons,Math.Notes USSR Acad.Sci:18:2291–300(1975).(English translation in Math.Notes18:2760–764)16.Zorich,A.:Flat surfaces.In:Cartier,P.,Julia,B.,Moussa,P.,Vanhove,P.(eds.)Frontiers in numbertheory,physics and geometry,Springer,Berlin(2006)123。